# Topology of manifolds

The branch of the theory of manifolds (cf. Manifold) concerned with the study of relations between different types of manifolds.

The most important types of finite-dimensional manifolds and relations between them are illustrated in (1).

 (1)

Here is the category of differentiable (smooth) manifolds; is the category of piecewise-linear (combinatorial) manifolds; is the category of topological manifolds that are polyhedra; is the category of topological manifolds admitting a topological decomposition into handles; is the category of Lipschitz manifolds (with Lipschitz transition mappings between local charts); is the category of topological manifolds (Hausdorff and with a countable base); is the category of polyhedral homology manifolds without boundary (polyhedra, the boundary of the star of each vertex of which has the homology of the sphere of corresponding dimension); is the category of generalized manifolds (finite-dimensional absolute neighbourhood retracts that are homology manifolds without boundary, i.e. with the property that for any point the group is isomorphic to the group ); is the category of Poincaré spaces (finite-dimensional absolute neighbourhood retracts for which there exists a number and an element such that when , and the mapping is an isomorphism for all ); and is the category of Poincaré polyhedra (the subcategory of the preceding category consisting of polyhedra).

The arrows of (1), apart from the 3 lower ones and the arrows , denote functors with the structure of forgetting functors. The arrow expresses Whitehead's theorem on the triangulability of smooth manifolds. In dimensions this arrow is reversible (an arbitrary -manifold is smoothable) but in dimensions there are non-smoothable -manifolds and even -manifolds that are homotopy inequivalent to any smooth manifold. The imbedding is also irreversible in the same strong sense (there exist polyhedral manifolds of dimension that are homotopy inequivalent to any -manifold). Here already for the sphere , , there exist triangulations in which it is not a -manifold.

The arrow expresses the fact that every -manifold has a handle decomposition.

The arrow expresses the theorem on the existence of a Lipschitz structure on an arbitrary -manifold.

The arrow is reversible if and irreversible if (an arbitrary topological manifold of dimension admits a handle decomposition and there exist four-dimensional topological manifolds for which this is not true).

Similarly, if the arrow is reversible (and moreover in a unique way).

The question on the reversibility of the arrow gives the classical unsolved problem on the triangulability of arbitrary topological manifolds.

The arrow is irreversible in the strong sense (there exist Poincaré polyhedra that are homotopy inequivalent to any homology manifold).

The arrow expresses a theorem on the homotopy equivalence of an arbitrary homology manifold of dimension to a topological manifold.

The arrow expresses the theorem on the homotopy equivalence of an arbitrary topological manifold to a polyhedron.

The imbedding expresses that an arbitrary topological manifold is an .

The similar question for the arrows has been solved using the theory of stable bundles (respectively, vector, piecewise-linear, topological, and spherical bundles), i.e. by examining the homotopy classes of mappings of a manifold into the corresponding classifying spaces BO, BPL, BTOP, BG.

There exist canonical composition mappings

 (2)

of which the homotopy fibres and the homotopy fibres of their compositions are denoted, respectively, by the symbols

For every manifold from a category , , , there exists a normal stable bundle, i.e. a canonical mapping from into the corresponding classifying space.

In the transition from a "narrow" category of manifolds to a "wider" one, for example, from smooth to piecewise-linear, the mapping is composed with the corresponding mappings (2). Hence, for example, for a PL-manifold there exists a smooth manifold PL-homeomorphic to it ( is said to be smoothable) only if the lifting problem (3), the homotopy obstruction to the solution of which lies in the groups , is solvable:

 (3)

Here it turns out that the solvability of (3) is not only necessary but also sufficient for the smoothability of a PL-manifold (and all non-equivalent smoothings are in bijective correspondence with the set of homotopy classes of mappings ).

By replacing by , the same holds for the smoothability of topological manifolds of dimension , and also (by replacing by ) for their -triangulations. The homotopy group is isomorphic to the group of classes of oriented diffeomorphic smooth manifolds obtained by glueing the boundaries of two -dimensional spheres. This group is finite for all (and is even trivial for ). Therefore, the number of non-equivalent smoothings of an arbitrary PL-manifold is finite and is bounded above by the number

The homotopy group vanishes, with one exception: . Thus, the existence of a -triangulation of a topological manifold of dimension is determined by the vanishing of a certain cohomology class , while the set of non-equivalent -triangulations of is in bijective correspondence with the group .

The group coincides with the group if and differs from for by the group . The number of non-equivalent smoothings of a topological manifold of dimension is finite and is bounded above by the number .

Similar results are not valid for Poincaré polyhedra.

 (4)

Of course, the existence of a lifting, for example, in (4) is a necessary condition for the existence of a PL-manifold homotopy equivalent to the Poincaré polyhedron , but, generally speaking, it ensures (for ) only the existence of a PL-manifold and a mapping of degree 1 such that . The transformation of this manifold into a manifold that is homotopy equivalent to requires the technique of surgery (reconstruction), initially developed by S.P. Novikov for the case when is a simply-connected smooth manifold of dimension . For simply-connected this technique has been generalized to the case under consideration. Thus, for a simply-connected Poincaré polyhedron a PL-manifold of dimension homotopy equivalent to it exists if and only a lifting (4) exists. The problem of the existence of topological or smooth manifolds that are homotopy equivalent to an (even simply-connected) Poincaré polyhedron is still more complicated.

#### References

 [1] S.P. Novikov, "On manifolds with free abelian group and their application" Izv. Akad. Nauk SSSR Ser. Mat. , 30 (1966) pp. 207–246 (In Russian) [2] J. Madsen, R.J. Milgram, "The classifying spaces for surgery and cobordism of manifolds" , Princeton Univ. Press (1979) [3] F. Latour, "Double suspension d'une sphere d'homologie [d'après R. Edwards]" , Sem. Bourbaki Exp. 515 , Lect. notes in math. , 710 , Springer (1979) pp. 169–186 [4] M.H. Freedman, "The topology of four-dimensional manifolds" J. Differential Geom. , 17 (1982) pp. 357–453 [5] F. Quinn, "Ends of maps III. Dimensions 4 and 5" J. Differential Geom. , 17 (1982) pp. 503–521 [6] R. Mandelbaum, "Four-dimensional topology: an introduction" Bull. Amer. Math. Soc. , 2 (1980) pp. 1–159 [7] R. Lashof, "The immersion approach to triangulation and smoothing" A. Liulevicius (ed.) , Algebraic Topology (Madison, 1970) , Proc. Symp. Pure Math. , 22 , Amer. Math. Soc. (1971) pp. 131–164 [8] R.D. Edwards, "Approximating certain cell-like maps by homeomorphisms" Notices Amer. Math. Soc. , 24 : 7 (1977) pp. A649 [9] F. Quinn, "The topological characterization of manifolds" Abstracts Amer. Math. Soc. , 1 : 7 (1980) pp. 613–614 [10] J.W. Cannon, "The recognition problem: what is a topological manifold" Bull. Amer. Math. Soc. , 84 : 5 (1978) pp. 832–866 [11] M. Spivak, "Spaces satisfying Poincaré duality" Topology , 6 (1967) pp. 77–101 [12] N.H. Kuiper, "A short history of triangulation and related matters" P.C. Baayen (ed.) D. van Dulst (ed.) J. Oosterhoff (ed.) , Bicentennial Congress Wisk. Genootschap (Amsterdam 1978) , Math. Centre Tracts , 100 , CWI (1979) pp. 61–79

It was found recently [a1] that the behaviour of smooth manifolds of dimension is radically different from those in dimensions . Among very numerous recent results one has:

i) There is a countably infinite family of smooth, compact, simply-connected -manifolds, all mutually homeomorphic but with distinct smooth structure.

ii) There is an uncountable family of smooth -manifolds, each homeomorphic to but with mutually distinct smooth structure.

iii) There are simply-connected smooth -manifolds which are -cobordant (cf. -cobordism) but not diffeomorphic.

For the lifting problem (3) see [a2][a3].